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CHAPTER-1 :- REAL NUMBERS
CHAPTER-1 :- REAL NUMBERS
Introduction
A real number is any number that can be represented on the number line.
Types of Real Numbers
Natural Numbers (N): 1, 2, 3, ...
Whole Numbers (W): 0, 1, 2, 3, ...
Integers (Z): ..., -2, -1, 0, 1, 2, ...
Rational Numbers (Q): Numbers that can be written as p/q, where q ≠ 0.
Irrational Numbers: Cannot be written as p/q (e.g., √2, √3, π).
Real Numbers (R): Rational + Irrational numbers.
2. Euclid's Division Lemma
Statement
For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:
a = bq + r, where 0 ≤ r < b
Example
Divide 25 by 4:
Quotient = 6
Remainder = 1
So: 25 = 4 × 6 + 1
3. Euclid's Division Algorithm
Steps
Divide the larger number by the smaller.
Find the remainder.
Replace the larger number with the smaller number and the smaller number with the remainder.
Continue until the remainder becomes 0.
The last non-zero remainder is the HCF.
Example
Find HCF of 96 and 72.
96 = 72 × 1 + 24
72 = 24 × 3 + 0
Therefore,
HCF = 24
4. Prime Numbers
A prime number has only two factors:
1
Itself
Examples: 2, 3, 5, 7, 11, 13...
2 is the only even prime number.
5. Composite Numbers
Numbers having more than two factors.
Examples: 4, 6, 8, 9, 10...
6. Fundamental Theorem of Arithmetic
Statement
Every composite number can be expressed as a product of prime numbers, and this factorization is unique (except for the order of factors).
Example: 60 = 2² × 3 × 5
7. Prime Factorization
Example: 84
84 = 2 × 42
42 = 2 × 21
21 = 3 × 7
Therefore:
84 = 2² × 3 × 7
8. HCF and LCM
HCF
Highest Common Factor
LCM
Least Common Multiple
Formula
HCF × LCM = Product of the two numbers
Example
12 and 18
HCF = 6
LCM = 36
Check:
6 × 36 = 216
12 × 18 = 216 ✓
9. Decimal Expansion
A rational number has:
Terminating Decimal
If denominator contains only prime factors 2 and/or 5.
Examples: 1/2 = 0.5
3/8 = 0.375
Non-Terminating Recurring Decimal
If denominator has any prime factor other than 2 or 5.
Example:
1/3 = 0.333...
10. Irrational Numbers
Cannot be expressed as p/q.
Examples:
√2
√5
π
Their decimal expansions are non-terminating and non-recurring.
11. Important Properties
Every prime number is a natural number.
Every natural number is a whole number.
Every whole number is an integer.
Every integer is a rational number.
Every rational number is a real number.
Irrational numbers are also real numbers.
12. Important Board Questions
1-Mark
State Euclid's Division Lemma.
Define prime number.
Define irrational number.
2-Mark
Find HCF using Euclid's Algorithm.
Write the prime factorization of a number.
4-Mark
Find HCF and LCM using prime factorization.
Explain the Fundamental Theorem of Arithmetic with an example.
Determine whether a rational number has a terminating or recurring decimal expansion.
13. Common Mistakes
❌ Forgetting that the remainder must satisfy 0 ≤ r < b.
❌ Calling 1 a prime number (it is neither prime nor composite).
❌ Thinking every decimal is irrational.
❌ Missing repeated prime factors in prime factorization.
POWERS & EXPONENTS
POWERS & EXPONENTS
•ᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• a¹⁄ⁿ = ⁿ√a
• a⁻ⁿ = 1/aⁿ
• (a/b)ᵐ = aᵐ/bᵐ
• a⁰ = 1
• (ab)ᵐ = aᵐ × bᵐ
• (aᵐ)ⁿ = aᵐⁿ
PROPORTIES OF LOGARITHMS
PROPORTIES OF LOGARITHMS
• log₁₀(ab) = log₁₀a + log₁₀b
• log₁₀(a/b) = log₁₀a − log₁₀b
• log₁₀(aᵐ) = m log₁₀a
• log₁₀(aᵐ⁄ⁿ) = (m/n) log₁₀a
• logₐ(aⁿ) = n
• logₐa = 1
• log₁₀10 = 1
• logₐ1 = 0
• If logₐm = x, then aˣ = m
• log a + log b + log c + log d = log(abcd)
• log₁₀√a = ½ log₁₀a
• logₐ(1/a) = −1
• log₁₀(a¹⁄²) = log₁₀√a
CHAPTER-2:- SETS
CHAPTER-2:- SETS
A Set is a well-defined collection of distinct objects.
The objects in a set are called elements or members.
Examples
A = {2, 4, 6, 8}
B = {a, e, i, o, u}
C = {January, February, March}
Not a Set:
Collection of beautiful flowers ❌ (because "beautiful" is not well-defined)
2. Representation of Sets
(A) Roster (Tabular) Form
Elements are listed inside curly brackets.
Example:
A = {1, 2, 3, 4, 5}
(B) Set Builder Form
A rule describes the elements.
Example:
A = {x | x is a natural number less than 6}
4. Types of Sets
• Empty (Null) Set (∅)
A set with no elements.
Example:
{Months with 32 days}
• Singleton Set
Contains only one element.
Example:
{0}
• Finite Set
Has a limited number of elements.
Example:
{2,4,6}
• Infinite Set
Has unlimited elements.
Example:
Natural Numbers
• Equal Sets
Contain exactly the same elements.
Example:
A={1,2,3}
B={3,2,1}
A = B
• Equivalent Sets
Contain the same number of elements, but elements may differ.
Example:
A={1,2,3}
B={a,b,c}
5. Universal Set
The set containing all elements under discussion.
Example:
U={1,2,3,4,5,6,7,8,9,10}
6. Subsets
If every element of A belongs to B, then
A ⊂ B
Example
A={2,4}
B={1,2,3,4,5}
Then
A⊂B
Proper Subset
A⊂B and A≠B
Improper Subset
Every set is a subset of itself.
7. Number of Subsets
If a set has n elements
Number of subsets
2ⁿ
Example
A={1,2,3}
Number of subsets
2³=8
8. Power Set
The collection of all subsets of a set.
Example
A={1,2}
Power Set
{∅,{1},{2},{1,2}
9. Union of Sets
All elements present in A or B.
Formula
A∪B
Example
A={1,2,3}
B={3,4,5}
A∪B={1,2,3,4,5}
10. Intersection
Common elements.
Formula
A∩B
Example
A={1,2,3}
B={3,4,5}
A∩B={3}
11. Difference of Sets
Elements in A but not in B.
Formula
A−B
Example
A={1,2,3}
B={3,4,5}
A−B={1,2}
12. Complement of a Set
Elements in Universal Set but not in A.
Formula
A′=U−A
Example
U={1,2,3,4,5}
A={2,4}
A′={1,3,5}
13. Cardinal Number
Number of elements in a set.
Notation
n(A)
Example
A={2,4,6}
n(A)=3
14. Important Formula
For two finite sets
n(A∪B)=n(A)+n(B)−n(A∩B)
Example
n(A)=20
n(B)=15
n(A∩B)=5
Then
n(A∪B)
=20+15−5
=30
15. Venn Diagram
A Venn Diagram is a pictorial representation of sets using circles.
Uses:
Show Union
Show Intersection
Show Complement
Solve counting problems
CHAPTER -3 :- POLYNOMIALS
CHAPTER -3 :- POLYNOMIALS
Definition
A polynomial is an algebraic expression made up of variables and constants connected by addition, subtraction, or multiplication. The exponent (power) of each variable must be a whole number (0, 1, 2, 3, ...).
Examples:
x + 2
3x^2 - 5x + 7
2x^3 + x - 1
Not Polynomials:
1/x (negative exponent)
sqrt(x) (fractional exponent)
x^-2
2. Parts of a Polynomial
Example: 5x^2 + 3x - 7
Terms: 5x^2, 3x, -7
Coefficient: The numerical value multiplied by the variable.
Examples: Coefficient of x^2 = 5 Coefficient of x = 3
Constant Term: A term without any variable.
Example: -7
Variable: A symbol representing an unknown value.
Example: x
3. Degree of a Polynomial
Definition: The highest power of the variable in a polynomial is called its degree.
Examples:
7 Degree = 0
x + 5 Degree = 1
x^2 + 2x + 1 Degree = 2
2x^3 - x Degree = 3
4. Types of Polynomials
Based on Degree
Constant Polynomial Degree = 0
Example: 5
Linear Polynomial Degree = 1
Example: 2x + 3
Quadratic Polynomial Degree = 2
Example: x^2 + 5x + 6
Cubic Polynomial Degree = 3
Example: x^3 + 2x^2 - 4
Based on Number of Terms
Monomial A polynomial having one term.
Example: 5x
Binomial A polynomial having two terms.
Example: x + 4
Trinomial A polynomial having three terms.
Example: x^2 + 3x + 2
5. Zero (Root) of a Polynomial
Definition: A zero of a polynomial is the value of the variable that makes the polynomial equal to zero.
Example:
p(x) = x - 4
Put x = 4
p(4) = 4 - 4 = 0
Therefore, 4 is the zero of the polynomial.
6. Relationship Between Zeroes and Coefficients
For a quadratic polynomial:
ax^2 + bx + c
Sum of Zeroes = -b/a
Product of Zeroes = c/a
Example:
x^2 - 5x + 6
a = 1 b = -5 c = 6
Sum of Zeroes
= -(-5)/1
= 5
Product of Zeroes
= 6/1
= 6
7. Remainder Theorem
Definition:
If a polynomial p(x) is divided by (x - a), then the remainder is equal to p(a).
Example:
p(x) = x^2 + 3x + 2
Divide by (x - 1)
Put x = 1
p(1)
= 1^2 + 3(1) + 2
= 1 + 3 + 2
= 6
Therefore,
Remainder = 6
8. Factor Theorem
Definition:
If p(a) = 0, then (x - a) is a factor of p(x).
Example:
p(x) = x^2 - 9
Put x = 3
p(3)
= 3^2 - 9
= 9 - 9
= 0
Therefore,
(x - 3) is a factor.
9. Factorization
Definition:
Factorization is the process of expressing a polynomial as the product of two or more simpler polynomials.
Example:
x^2 + 5x + 6
= (x + 2)(x + 3)
Verification:
(x + 2)(x + 3)
= x^2 + 3x + 2x + 6
= x^2 + 5x + 61.
Example: 5x² + 3x − 7Terms: 5x², 3x, −7
Coefficients: 5, 3
Constant te
CHAPTER-4:- PAIR OF LINEAR EQUATION IN TWO VARIABLES
CHAPTER-4:- PAIR OF LINEAR EQUATION IN TWO VARIABLES
Definition
A pair of linear equations in two variables consists of two linear equations with the same two variables (usually x and y). The solution is the value of x and y that satisfies both equations.
General Form:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
where a1, b1, c1, a2, b2, and c2 are constants.
2. Linear Equation
Definition: A linear equation is an equation in which the highest power of the variable is 1.
Examples:
2x + 3y = 7
x - y = 5
Not a Linear Equation:
x^2 + y = 6
xy = 10
3. Solution of a Pair of Linear Equations
Definition: The values of x and y that satisfy both equations are called the solution.
Example:
x + y = 7
x - y = 3
Add the equations:
2x = 10
x = 5
Substitute x = 5 into x + y = 7:
5 + y = 7
y = 2
Solution: (5, 2)
4. Methods of Solving
(i) Graphical Method
Draw the graphs of both equations.
The point where the two lines intersect is the solution.
(ii) Substitution Method
Find one variable from one equation.
Substitute it into the other equation.
Find the second variable.
(iii) Elimination Method
Make the coefficients of one variable equal.
Add or subtract the equations to eliminate one variable.
Find the remaining variable.
(iv) Cross Multiplication Method
For
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1)
5. Types of Solutions
(i) Unique Solution
The two lines intersect at one point.
Condition:
a1/a2 ≠ b1/b2
(ii) No Solution
The two lines are parallel.
Condition:
a1/a2 = b1/b2 ≠ c1/c2
(iii) Infinitely Many Solutions
The two lines coincide.
Condition:
a1/a2 = b1/b2 = c1/c2
6. Graphical Representation
Intersecting lines → One solution
Parallel lines → No solution
Coincident lines → Infinitely many solutions
CHAPTER-5 :- QUADRATIC EQUATIONS
CHAPTER-5 :- QUADRATIC EQUATIONS
Definition
A quadratic equation is an equation of degree 2 (highest power of the variable is 2).
General Form:
ax^2 + bx + c = 0
where:
a, b and c are real numbers.
a ≠ 0
Examples:
x^2 + 5x + 6 = 0
2x^2 - 3x + 1 = 0
Not a Quadratic Equation:
x + 5 = 0 (Linear)
x^3 + 2x = 0 (Cubic)
2. Standard Form
Every quadratic equation should be written in the form:
ax^2 + bx + c = 0
Example:
2x(x + 3) = 20
2x^2 + 6x = 20
2x^2 + 6x - 20 = 0
(Standard Form)
3. Roots (Zeroes) of a Quadratic Equation
Definition:
The values of x that satisfy the quadratic equation are called roots (or zeroes).
Example:
x^2 - 5x + 6 = 0
Roots are:
x = 2 and x = 3
because
2^2 - 5(2) + 6 = 0
3^2 - 5(3) + 6 = 0
4. Methods of Solving Quadratic Equations
(i) Factorization Method
Split the middle term and factorize.
Example:
x^2 + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = -2 or x = -3
(ii) Completing the Square Method
Convert the equation into a perfect square and solve.
Example:
x^2 + 6x + 5 = 0
x^2 + 6x = -5
Add 9 to both sides.
(x + 3)^2 = 4
x + 3 = ±2
x = -1 or -5
(iii) Quadratic Formula
Formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
This method works for every quadratic equation.
5. Discriminant
Definition:
The expression inside the square root is called the Discriminant.
D = b^2 - 4ac
6. Nature of Roots
If D > 0
Two distinct real roots.
If D = 0
Two equal real roots.
If D < 0
No real roots.
7. Sum and Product of Roots
For
ax^2 + bx + c = 0
Sum of roots = -b/a
Product of roots = c/a
Example:
x^2 - 7x + 12 = 0
Sum = 7
Product = 12
8. Forming a Quadratic Equation
If the roots are α and β,
Equation:
x^2 - (α + β)x + αβ = 0
Example:
Roots = 2 and 5
Sum = 7
Product = 10
Equation:
x^2 - 7x + 10 = 0
CHAPTER-6:- ARITHMETIC PROGRESSIONS
CHAPTER-6:- ARITHMETIC PROGRESSIONS
Definition
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same. This fixed difference is called the Common Difference (d).
Examples:
2, 5, 8, 11, 14, ...
10, 7, 4, 1, ...
Not an AP:
2, 4, 8, 16, ... (Difference is not constant.)
2. Terms Used in AP
First Term (a): The first number of the sequence.
Common Difference (d): Difference between consecutive terms.
Formula:
d = Second Term - First Term
Example:
5, 8, 11, 14...
a = 5
d = 8 - 5 = 3
3. General Form of AP
a, a + d, a + 2d, a + 3d, ...
Example:
4, 7, 10, 13...
a = 4
d = 3
4. nth Term of an AP
Definition: The nth term is the term at the nth position of an AP.
Formula:
an = a + (n - 1)d
Example:
Find the 10th term of
2, 5, 8, 11...
Given:
a = 2
d = 3
n = 10
an = 2 + (10 - 1) × 3
= 2 + 27
= 29
Therefore, the 10th term is 29.
5. Last Term of an AP
Formula:
l = a + (n - 1)d
where l = last term.
6. Sum of First n Terms
Definition: The total of the first n terms of an AP.
Formula:
Sn = n/2 [2a + (n - 1)d]
OR
Sn = n(a + l)/2
where l = last term.
Example:
Find the sum of the first 10 terms of
2, 5, 8, 11...
a = 2
d = 3
n = 10
Sn = 10/2 [2(2) + 9 × 3]
= 5(4 + 27)
= 5 × 31
= 155
7. Increasing and Decreasing AP
Increasing AP: If d is positive.
Example:
2, 5, 8, 11...
Decreasing AP: If d is negative.
Example:
20, 17, 14, 11...
CHAPTER-7:- COORDINATE GEOMETRY
CHAPTER-7:- COORDINATE GEOMETRY
Introduction
Coordinate Geometry is the branch of mathematics that helps us locate the position of a point on a plane using two perpendicular lines called the X-axis and Y-axis.
The point where the X-axis and Y-axis intersect is called the Origin (O).
Coordinates of the origin are (0, 0).
2. Cartesian Plane
The Cartesian Plane consists of:
X-axis (Horizontal axis)
Y-axis (Vertical axis)
Origin (0,0)
A point is represented as (x, y), where:
x = x-coordinate (Abscissa)
y = y-coordinate (Ordinate)
Example: Point A(3,4)
x-coordinate = 3
y-coordinate = 4
3. Quadrants
The Cartesian Plane is divided into four quadrants.
First Quadrant (I): x is positive, y is positive Example: (3,2)
Second Quadrant (II): x is negative, y is positive Example: (-3,2)
Third Quadrant (III): x is negative, y is negative Example: (-3,-2)
Fourth Quadrant (IV): x is positive, y is negative Example: (3,-2)
4. Distance Formula
Definition: The distance formula is used to find the distance between two points.
Formula:
Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
Example:
Find the distance between A(2,3) and B(6,6).
Distance
= sqrt[(6 - 2)^2 + (6 - 3)^2]
= sqrt[4^2 + 3^2]
= sqrt[16 + 9]
= sqrt25
= 5 units
5. Midpoint Formula
Definition: The midpoint is the point exactly halfway between two points.
Formula:
Midpoint = ((x1 + x2)/2 , (y1 + y2)/2)
Example:
Find the midpoint of A(2,4) and B(6,8).
Midpoint
= ((2 + 6)/2 , (4 + 8)/2)
= (4,6)
6. Section Formula
Definition: The section formula is used to find a point dividing a line segment in a given ratio.
Formula:
((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n))
where the ratio is m : n.
CHAPTER-8:- SIMILAR TRIANGLES
CHAPTER-8:- SIMILAR TRIANGLES
Introduction
Two triangles are said to be similar if they have the same shape but not necessarily the same size.
In similar triangles:
Corresponding angles are equal.
Corresponding sides are proportional.
Symbol: △ABC ~ △DEF
2. Conditions for Similarity
(i) AAA (Angle-Angle-Angle) Similarity
If the three corresponding angles of two triangles are equal, then the triangles are similar.
Example: ∠A = ∠D ∠B = ∠E ∠C = ∠F
Then, △ABC ~ △DEF
(ii) SAS (Side-Angle-Side) Similarity
If two sides are proportional and the included angle is equal, then the triangles are similar.
Example:
AB/DE = AC/DF
and
∠A = ∠D
Therefore,
△ABC ~ △DEF
(iii) SSS (Side-Side-Side) Similarity
If the three corresponding sides of two triangles are proportional, then the triangles are similar.
Example:
AB/DE = BC/EF = AC/DF
Therefore,
△ABC ~ △DEF
3. Properties of Similar Triangles
Corresponding angles are equal.
Corresponding sides are proportional.
Ratio of their perimeters is equal to the ratio of corresponding sides.
Ratio of their areas is equal to the square of the ratio of corresponding sides.
Example:
If AB/DE = 2/3
Then,
Area of △ABC / Area of △DEF = 4/9
4. Basic Proportionality Theorem (BPT)
Statement
If a line is drawn parallel to one side of a triangle to intersect the other two sides, then it divides those two sides in the same ratio.
Example:
In △ABC,
If DE || BC,
then
AD/DB = AE/EC
This theorem is also called Thales' Theorem.
5. Converse of BPT
Statement
If a line divides any two sides of a triangle in the same ratio, then that line is parallel to the third side.
Example:
If
AD/DB = AE/EC
Then
DE || BC
6. Pythagoras Theorem
Statement
In a right-angled triangle,
(Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2
Formula:
c^2 = a^2 + b^2
Example:
Base = 6 cm
Height = 8 cm
Hypotenuse = ?
c^2 = 6^2 + 8^2
= 36 + 64
= 100
c = 10 cm
7. Converse of Pythagoras Theorem
Statement
If in a triangle,
c^2 = a^2 + b^2,
then the triangle is a right-angled triangle.
CHAPTER-9:- TRIGONOMETRY
CHAPTER-9:- TRIGONOMETRY
Introduction
Definition: Trigonometry is the branch of mathematics that deals with the relationship between the angles and sides of a right-angled triangle.
The word trigonometry means:
Tri = Three
Gon = Sides
Metron = Measurement
2. Parts of a Right-Angled Triangle
A right-angled triangle has:
Hypotenuse:
The side opposite to the right angle.
It is the longest side.
perpendicular:
The side opposite to the given angle.
Base:
The remaining side.
3. Trigonometric Ratios
For an angle θ:
1. Sine (sin)
sin θ = Perpendicular / Hypotenuse
2. Cosine (cos)
cos θ = Base / Hypotenuse
3. Tangent (tan)
tan θ = Perpendicular / Base
4. Cosecant (cosec)
cosec θ = Hypotenuse / Perpendicular
cosec θ = 1/sin θ
5. Secant (sec)
sec θ = Hypotenuse / Base
sec θ = 1/cos θ
6. Cotangent (cot)
cot θ = Base / Perpendicular
cot θ = 1/tan θ
4. Trigonometric Identities
Identity 1:
sin²θ + cos²θ = 1
Identity 2:
1 + tan²θ = sec²θ
Identity 3:
1 + cot²θ = cosec²θ
5. Trigonometric Ratios of Standard Angles
Angles:
0°, 30°, 45°, 60°, 90°
sin values:
0° = 0
30° = 1/2
45° = 1/sqrt(2)
60° = sqrt(3)/2
90° = 1
cos values:
0° = 1
30° = sqrt(3)/2
45° = 1/sqrt(2)
60° = 1/2
90° = 0
tan values:
0° = 0
30° = 1/sqrt(3)
45° = 1
60° = sqrt(3)
90° = Not defined
6. Important Relations
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
sec θ = 1/cos θ
cosec θ = 1/sin θ
7. Finding Trigonometric Ratios
Example:
If sin θ = 3/5, find cos θ.
sin θ = Perpendicular/Hypotenuse
Perpendicular = 3
Hypotenuse = 5
Using Pythagoras:
Base² = 5² - 3²
= 25 - 9
= 16
Base = 4
Therefore:
cos θ = Base/Hypotenuse
= 4/5
8. Complementary Angles
If two angles add up to 90°, they are complementary.
Relations:
sin(90° - θ) = cos θ
cos(90° - θ) = sin θ
tan(90° - θ) = cot θ
cot(90° - θ) = tan θ
sec(90° - θ) = cosec θ
cosec(90° - θ) = sec θ
9. Applications of Trigonometry
Trigonometry is used to find:
Heights of buildings
Heights of trees
Distance from objects
Angles of elevation and depression
CHAPTER-10:- APPLICATIONS OF TRIGONOMETRY
CHAPTER-10:- APPLICATIONS OF TRIGONOMETRY
Introduction
Definition: Applications of Trigonometry is the use of trigonometric ratios to find the height or distance of objects without measuring them directly.
Examples:
Height of a building
Height of a tree
Width of a river
Distance of a ship from a lighthouse
2. Line of Sight
Definition: The straight line drawn from the observer's eye to the object is called the line of sight.
3. Angle of Elevation
Definition: The angle formed between the horizontal line and the line of sight when the object is above the observer's eye level is called the angle of elevation.
Examples:
Looking at the top of a building.
Looking at the top of a tower.
4. Angle of Depression
Definition: The angle formed between the horizontal line and the line of sight when the object is below the observer's eye level is called the angle of depression.
Examples:
Looking down from a bridge.
Looking down from the top of a building.
5. Trigonometric Ratios Used
sin θ = Perpendicular / Hypotenuse
cos θ = Base / Hypotenuse
tan θ = Perpendicular / Base
Most board questions use tan θ.
6. Solving Height and Distance Problems
Step 1
Draw a right-angled triangle.
Step 2
Write the given measurements.
Step 3
Choose the correct trigonometric ratio.
Step 4
Substitute the values.
Step 5
Find the required height or distance
7. Example
A tree casts a shadow of 10 m. The angle of elevation of the Sun is 45°. Find the height of the tree.
Given:
Shadow = 10 m
Angle = 45°
Using
tan 45° = Height / Shadow
1 = Height / 10
Height = 10 m
Answer: Height of the tree = 10 m
8. Important Standard Values
sin 30° = 1/2
cos 30° = sqrt(3)/2
tan 30° = 1/sqrt(3)
sin 45° = 1/sqrt(2)
cos 45° = 1/sqrt(2)
tan 45° = 1
sin 60° = sqrt(3)/2
cos 60° = 1/2
tan 60° = sqrt(3)
9. Applications in Daily Life
Applications of Trigonometry are used in:
Surveying
Navigation
Architecture
Engineering
Construction of buildings and bridges
Measuring heights and distances
10. Important Formulae
tan θ = Height / Distance
Height = Distance × tan θ
Distance = Height / tan θ
CHAPTER-11:- TANGENTS AND SECANTS TO A CIRCLE
CHAPTER-11:- TANGENTS AND SECANTS TO A CIRCLE
Introduction
Definition: A circle is the set of all points in a plane that are at the same distance from a fixed point.
The fixed point is called the centre of the circle.
2. Important Terms
Centre
The fixed point inside the circle.
Radius (r)
A line segment joining the centre to any point on the circle.
Diameter (d)
A line segment passing through the centre and joining two points on the circle.
Formula: d = 2r
Chord
A line segment joining any two points on the circle.
Arc
A part of the circumference of a circle.
Circumference
The boundary or outer edge of a circle.
Tangent
A line that touches the circle at exactly one point.
Point of Contact
The point where the tangent touches the circle.
Secant
A line that cuts the circle at two points.
3. Tangent to a Circle
Definition: A tangent is a straight line that touches the circle at only one point.
Important Property
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Radius ⟂ Tangent
4. Theorem 1
Statement
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Example:
If OA is the radius and AB is the tangent,
then
OA ⟂ AB
Angle OAB = 90°
5. Theorem 2
Statement
The lengths of tangents drawn from an external point to a circle are equal.
If PA and PB are tangents from point P,
then
PA = PB
6. Important Properties
A circle has an infinite number of tangents.
A tangent touches the circle at only one point.
A radius drawn to the point of contact is always perpendicular to the tangent.
Tangents drawn from the same external point are equal in length.
7. Applications of Circles
Circles are used in:
Wheels
Clocks
Coins
Circular parks
Ferris wheels
Engineering and architecture
8. Important Formulae
Diameter = 2 × Radius
Radius = Diameter / 2
Circumference = 2πr
Area = πr^2
CHAPTER-12:- MENSURATION
CHAPTER-12:- MENSURATION
Introduction
Definition: Surface Area and Volume is the branch of mathematics that deals with measuring the outer surface and the space occupied by three-dimensional (3D) objects.
Examples of 3D shapes:
Cube
Cuboid
Cylinder
Cone
Sphere
Hemisphere
2. Surface Area
Definition: The total area covered by the outer surface of a solid object is called its Surface Area.
Types of Surface Area
Curved Surface Area (CSA): The area of only the curved part of a solid.
Total Surface Area (TSA): The area of the entire outer surface, including the base(s).
3. Volume
Definition: The amount of space occupied by a three-dimensional object is called its Volume.
Units:
cm^3
m^3
4. Cube
A cube has 6 equal square faces.
Formulae:
Side = a
LSA = 4a^2
TSA = 6a^2
Volume = a^3
5. Cuboid
A cuboid has length (l), breadth (b), and height (h).
Formulae:
LSA = 2h(l + b)
TSA = 2(lb + bh + lh)
Volume = l × b × h
6. Cylinder
A cylinder has two circular bases and one curved surface.
Formulae:
Radius = r
Height = h
CSA = 2πrh
TSA = 2πr(r + h)
Volume = πr^2h
7. Cone
A cone has one circular base and one curved surface.
Slant Height:
l = sqrt(r^2 + h^2)
Formulae:
CSA = πrl
TSA = πr(l + r)
Volume = (1/3)πr^2h
8. Sphere
A sphere has only one curved surface.
Formulae:
Surface Area = 4πr^2
Volume = (4/3)πr^3
9. Hemisphere
A hemisphere is half of a sphere.
Formulae:
CSA = 2πr^2
TSA = 3πr^2
Volume = (2/3)πr^3
10. Unit Conversions
1 m = 100 cm
1 m^2 = 10,000 cm^2
1 m^3 = 1,000,000 cm^3
11. Applications
Surface areas and volumes are used in:
Water tanks
Pipes
Buildings
Storage containers
Engineering
Packaging
12. Important Formulae
Cube:
TSA = 6a^2
Volume = a^3
Cuboid:
TSA = 2(lb + bh + lh)
Volume = l × b × h
Cylinder:
CSA = 2πrh
TSA = 2πr(r + h)
Volume = πr^2h
Cone:
l = sqrt(r^2 + h^2)
CSA = πrl
TSA = πr(l + r)
Volume = (1/3)πr^2h
Sphere:
Surface Area = 4πr^2
Volume = (4/3)πr^3
Hemisphere:
CSA = 2πr^2
TSA = 3πr^2
Volume = sqrt(r^2 + h^2)
CHAPTER-13:- STATISTICS
CHAPTER-13:- STATISTICS
Introduction
Definition: Statistics is the branch of mathematics that deals with the collection, organization, presentation, analysis, and interpretation of numerical data.
It helps us understand data and make comparisons.
Examples:
Marks of students
Population of cities
Rainfall data
Monthly income
2. Important Terms
Data
A collection of facts or numerical information.
Observation
Each value in the data is called an observation.
Frequency
The number of times an observation occurs.
Frequency Distribution
A table showing observations and their frequencies.
Class Interval
The range between the lower limit and upper limit.
Example: 10–20
3. Mean (Average)
Definition: The mean is the sum of all observations divided by the total number of observations.
Formula (Ungrouped Data)
Mean = Sum of observations / Number of observations
Example:
Data: 5, 8, 10, 12, 15
Mean = (5 + 8 + 10 + 12 + 15) / 5
= 50 / 5
= 10
4. Median
Definition: The median is the middle value of the data when arranged in ascending or descending order.
Steps
Arrange the data.
Find the middle value.
Example:
2, 5, 7, 9, 11
Median = 7
5. Mode
Definition: The mode is the observation that occurs most frequently.
Example:
4, 5, 5, 6, 8
Mode = 5
6. Cumulative Frequency
Definition: The running total of frequencies is called cumulative frequency.
It is mainly used to find the median of grouped data.
7. Graphical Representation
Statistics can be represented using:
Bar Graph
Histogram
Frequency Polygon
Ogive (Cumulative Frequency Curve)
Types of Mean
1. Mean of Ungrouped Data
Used when observations are given individually.
Formula: Mean = Sum of observations / Number of observations
Example: Data: 5, 10, 15, 20
Mean = (5 + 10 + 15 + 20) / 4 = 12.5
2. Mean of Grouped Data
(a) Direct Method
Formula:
Mean = (Σfi × xi) / Σfi
where:
fi = Frequency
xi = Class mark
(b) Assumed Mean Method
Used when calculations are lengthy.
Formula:
Mean = A + (Σfi × di / Σfi)
where:
A = Assumed Mean
di = xi − A
(c) Step-Deviation Method
Used when class intervals are equal and calculations can be simplified.
Formula:
Mean = A + [(Σfi × ui) / Σfi] × h
where:
ui = (xi − A) / h
h = Class size
Types of Median
1. Median of Ungrouped Data
Arrange the data and find the middle value.
2. Median of Grouped Data
Formula:
Median = l + [(N/2 − cf) / f] × h
where:
l = Lower boundary of median class
N = Total frequency
cf = Cumulative frequency before median class
f = Frequency of median class
h = Class size
Types of Mode
1. Mode of Ungrouped Data
The value occurring most frequently.
2. Mode of Grouped Data
Formula:
Mode = l + [(f1 − f0) / (2f1 − f0 − f2)] × h
where:
l = Lower boundary of modal class
f1 = Frequency of modal class
f0 = Frequency of class before modal class
f2 = Frequency of class after modal class
h = Class size
CHAPTER-14:- PROBABILITY
CHAPTER-14:- PROBABILITY
Introduction
Definition: Probability is the branch of mathematics that measures the chance of an event occurring.
The value of probability always lies between 0 and 1.
Examples:
Tossing a coin
Rolling a dice
Drawing a card
Choosing a ball from a bag
2. Experiment
Definition: An experiment is an action or process that produces one or more outcomes.
Examples:
Tossing a coin
Rolling a dice
Drawing a card
3. Trial
Definition: Each repetition of an experiment is called a trial.
Example: Tossing a coin once is one trial.
4. Outcome
Definition: A possible result of an experiment is called an outcome.
Examples:
Coin → Head (H), Tail (T)
Dice → 1, 2, 3, 4, 5, 6
5. Sample Space
Definition: The set of all possible outcomes of an experiment is called the sample space.
Symbol: S
Examples:
Coin: S = {H, T}
Dice: S = {1, 2, 3, 4, 5, 6}
6. Event
Definition: An event is a collection of one or more outcomes from the sample space.
Example:
Getting an even number on a dice.
Event = {2, 4, 6}
7. Types of Events
(i) Simple Event
An event containing only one outcome.
Example: Getting 3 on a dice.
(ii) Compound Event
An event containing more than one outcome.
Example: Getting an even number on a dice = {2, 4, 6}
(iii) Certain Event
An event that always occurs.
Probability = 1
Example: Getting a number less than 7 on a dice.
(iv) Impossible Event
An event that never occurs.
Probability = 0
Example: Getting 8 on a dice.
8. Probability Formula
Definition: Probability of an event is the ratio of the number of favourable outcomes to the total number of possible outcomes.
Formula:
P(E) = Number of favourable outcomes / Total number of outcomes
9. Properties of Probability
0 ≤ P(E) ≤ 1
Probability of a certain event = 1
Probability of an impossible event = 0
Sum of probabilities of all outcomes = 1
10. Examples
Example 1
A coin is tossed.
Find the probability of getting a Head.
Sample Space:
S = {H, T}
Favourable outcomes = 1
Total outcomes = 2
P(H) = 1/2
Example 2
A dice is rolled.
Find the probability of getting an even number.
Sample Space:
S = {1, 2, 3, 4, 5, 6}
Even numbers = {2, 4, 6}
Favourable outcomes = 3
Total outcomes = 6
P(Even) = 3/6 = 1/2
11. Applications of Probability
Probability is used in:
Weather forecasting
Games
Sports
Insurance
Medical research
Business and finance
12. Important Formulae
Probability:
P(E) = Favourable Outcomes / Total Outcomes
Impossible Event:
P(E) = 0
Certain Event:
P(E) = 1
Complementary Event:
P(Not E) = 1 - P(E)Introduction
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